The usual cause of that problem is a steel pipe or fitting in the line that is rusting. It would be unusual for the same one to affect both the sink and tub, so I will assume that there are two different pipes installed.

: I have noticed that when I don't use my Bathroom sink for a few hours, the water that comes out at first has a rusty tint to it.: If I run the tap a few seconds it goes away. The same happens in the bathtub. Does anyone have any ideas what could be causing this, and how to fix it?: BTW the faucets are brass, not chrome.